Convertible motor vehicles, commonly referred to as "convertibles," have tops which typically consist of a flexible cover and a supporting frame. To put the top down, the frame is retracted, often with considerable difficulty, to a position at the rear of the vehicle where it is not completely out of sight, thus detracting from the esthetic appearance of the vehicle as well as at least partially obstructing the driver's vision. The flexible covers of these tops are of one piece construction and are either completely deployed or completely retracted and stored. What is needed is a top for a convertible which is easy to deploy and retract and which may be stored completely out of sight when not in use, and also which is made in two parts so that one or both of the parts may be retracted and stored depending on the wishes of the operator.